Question

a family has two children. if b represents a boy and g represents a girl, the set of outcomes for the possible genders of the children is ( s = {bb, bg, gb, gg} ), with the oldest child listed first in each pair. let ( x ) represent the number of times ( g ) occurs. which of the following is the probability distribution, ( p(x) )?

Explanation:

Step1: Determine possible values of \( X \)

The number of girls (\( X \)) in two children can be 0, 1, or 2. So we analyze each case.

Step2: Calculate probabilities for each \( X \)

• Case \( X = 0 \) (no girls): Only outcome is \( BB \). Total outcomes: 4 (\( BB, BG, GB, GG \)). So \( P(X = 0) = \frac{1}{4} = 0.25 \).
• Case \( X = 1 \) (one girl): Outcomes are \( BG \) and \( GB \). So \( P(X = 1) = \frac{2}{4} = 0.5 \).
• Case \( X = 2 \) (two girls): Only outcome is \( GG \). So \( P(X = 2) = \frac{1}{4} = 0.25 \).

Step3: Match with the tables

The first table has \( P(0)=0.25 \), \( P(1)=0.5 \), \( P(2)=0.25 \), which matches our calculations.

Answer:

The first table (with \( X=0 \) having \( P(X)=0.25 \), \( X=1 \) having \( P(X)=0.5 \), \( X=2 \) having \( P(X)=0.25 \))